CORDIC算法原理详解及其Verilog实现
CORDIC算法原理详解及其Verilog实现
- CORDIC简介
- 算法实现
- 1. 已 知 坐 标 ( x , y ) ， 求 其 向 量 对 应 的 相 角 θ （ 反 正 切 ） 和 模 值 \color{#F00}{1.已知坐标(x,y)，求其向量对应的相角θ（反正切）和模值} 1.已知坐标(x,y)，求其向量对应的相角θ（反正切）和模值
- `1.1 MATLAB代码`
- `1.2 Verilog代码`
- `1.3 testbench A`
- 2. 已 知 角 度 θ ， 求 正 弦 s i n θ 和 余 弦 c o s θ \color{#F00}{2.已知角度θ，求正弦sinθ和余弦cosθ} 2.已知角度θ，求正弦sinθ和余弦cosθ
- `2.1 MATLAB代码`
- `2.2 Verilog代码`
- `2.3 Testbench B`
本文的verilog代码
链接：https://pan.baidu.com/s/1GGbRjxO5CxoIODQAg1l6Lw
提取码：jo0h
CORDIC简介
CORDIC（Coordinate Rotation Digital Computer）坐标旋转数字计算机，是数学与计算机技术交叉产生的一种机器算法，用于解决计算机的数学计算问题。发展到现在，CORDIC算法及其扩展算法大致有三种计算模式：圆周旋转模式、线性旋转模式和双曲线旋转模式，分别用来实现不同的数学运算。本文介绍圆周旋转模式下的CORDIC算法原理及实现过程，另两种模式将分期介绍。
简单来讲，CORDIC利用近似逼近的思想，将计算机中三角函数、开根号、求对数等复杂运算，转化为简单的加减和移位操作。
1）算法描述
如图，XY平面中一点 ( x 1 , y 1 ) (x_{1},y_{1}) (x1,y1)经圆周旋转θ角度，得到点 ( x 2 , y 2 ) (x_{2},y_{2}) (x2,y2)。经简单的三角函数关系，可得到：
x 2 = x 1 c o s θ − y 1 s i n θ x_{2} = x_{1}cos\theta - y_{1}sin\theta x2=x1cosθ−y1sinθ
y 2 = x 1 s i n θ + y 1 c o s θ y_{2} = x_{1}sin\theta + y_{1}cos\theta y2=x1sinθ+y1cosθ
通过提出因数cosθ，可得：
x 2 = x 1 c o s θ − y 1 s i n θ = c o s θ ( x 1 − y 1 t a n θ ) x_{2} = x_{1}cos\theta - y_{1}sin\theta = cos\theta(x_{1}-y_{1}tan\theta) x2=x1cosθ−y1sinθ=cosθ(x1−y1tanθ)
y 2 = x 1 s i n θ + y 1 c o s θ = c o s θ ( y 1 + x 1 t a n θ ) y_{2} = x_{1}sin\theta + y_{1}cos\theta = cos\theta(y_{1} + x_{1}tan\theta) y2=x1sinθ+y1cosθ=cosθ(y1+x1tanθ)
如果不考虑cosθ，得到“伪旋转”方程：
x 2 ^ = c o s θ ( x 1 − y 1 t a n θ ) = x 1 − y 1 t a n θ \hat{x_{2}} = \cancel{cos\theta}(x_{1}-y_{1}tan\theta)=x_{1}-y_{1}tan\theta x2^=cosθ (x1−y1tanθ)=x1−y1tanθ
y 2 ^ = c o s θ ( y 1 + x 1 t a n θ ) = y 1 + x 1 t a n θ \hat{y_{2}} = \cancel{cos\theta}(y_{1} + x_{1}tan\theta)=y_{1} + x_{1}tan\theta y2^=cosθ (y1+x1tanθ)=y1+x1tanθ
伪旋转仅实现了正确的角度旋转，但向量模值变为原来的1/cosθ。
2）CORDIC方法
若使得 t a n θ = 2 − i tanθ = 2^{-i} tanθ=2−i， i i i为自然数，则上式变为：
x 2 ^ = x 1 − y 1 t a n θ = x 1 − y 1 2 − i \hat{x_{2}} = x_{1} - y_{1}tan\theta = x_{1}-y_{1}2^{-i} x2^=x1−y1tanθ=x1−y12−i
y 2 ^ = y 1 + x 1 t a n θ = y 1 + x 1 2 − i \hat{y_{2}} = y_{1} + x_{1}tan\theta = y_{1} + x_{1}2^{-i} y2^=y1+x1tanθ=y1+x12−i
注意!上式已经仅剩加减和移位运算。
如果我们把所有 t a n θ = 2 − i tanθ = 2^{-i} tanθ=2−i对应的相关角度和正切值制成一张表，如下所示：
可以确定的是，任意旋转角度θ，都可以由上表中的大小不同 θ i θ_{i} θi进行多次旋转得到。CORDIC正是利用这一点，将θ角度的旋转分解为从大到小、逐次逼近真实旋转角度的一组旋转，而这些旋转的实现又都可以由加减和移位运算来完成。θi可以预先制成表格，供计算机查找使用。
算法实现
1. 已 知 坐 标 ( x , y ) ， 求 其 向 量 对 应 的 相 角 θ （ 反 正 切 ） 和 模 值 \color{#F00}{1.已知坐标(x,y)，求其向量对应的相角θ（反正切）和模值} 1.已知坐标(x,y)，求其向量对应的相角θ（反正切）和模值
思想:把原向量逐次向X正轴进行伪旋转逼近，整个过程的累计旋转角度即为θ，而旋转后的x坐标补偿模值增益K后即为原向量模值。
每次迭代的方程为：
x ( i + 1 ) = x ( i ) − d i ( 2 − i y ( i ) ) x^{(i+1)} = x^{(i)}-d_{i}(2^{-i}y^{(i)}) x(i+1)=x(i)−di(2−iy(i))
y ( i + 1 ) = y ( i ) − d i ( 2 − i x ( i ) ) y^{(i+1)} = y^{(i)}-d_{i}(2^{-i}x^{(i)}) y(i+1)=y(i)−di(2−ix(i))
模值总增益K为：
z ( i + 1 ) = z ( i ) − d i θ i z^{(i+1)} = z^{(i)}-d_{i}\theta^{i} z(i+1)=z(i)−diθi
其中d决定旋转方向为逆时针还是顺时针，z为角度累加值。
注意！查找表中所有θi的总和约为99度，因此能实现计算的角度在-99~99度之间。实际算法实现时，我们可预先判断坐标象限，人为将其转化为第一象限内的计算，并在事后补偿为真实值。
1.1 MATLAB代码
%% ***********************************************************************************
% 已知坐标，用cordic算法计算相角和幅值。基本公式如下：
% x(k+1) = x(k) - d(k)*y(k)*2^(-k)
% y(k+1) = y(k) + d(k)*x(k)*2^(-k)
% z(k) = z(k) - d(k)*actan(2^(-k))
%% ***********************************************************************************
clear;close all;clc;
% 初始化----------------------------------------
N = 16; %迭代次数
tan_table = 2.^-(0 : N-1);
angle_LUT = atan(tan_table);
K = 1;
for k = 0 : N-1
K = K*(1/sqrt(1 + 2^(-2*k)));
end
x = 3;
y = sqrt(3);
angle_accumulate = 0;
% cordic算法计算-------------------------------
if (x==0 && y==0)
radian_out = 0;
amplitude_out = 0;
else % 先做象限判断，得到相位补偿值
if (x > 0)
phase_shift = 0;
elseif (y < 0)
phase_shift = -pi;
else
phase_shift = pi;
end
for k = 0 : N-1 % 迭代开始
x_temp = x;
if (y < 0) % d(k)=1，逆时针旋转
x = x_temp - y*2^(-k);
y = y + x_temp*2^(-k);
angle_accumulate = angle_accumulate - angle_LUT(k+1);
else % d(k)=-1，顺时针旋转
x = x_temp + y*2^(-k);
y = y - x_temp*2^(-k);
angle_accumulate = angle_accumulate + angle_LUT(k+1);
end
radian_out = angle_accumulate + phase_shift; %弧度输出
end
amplitude_out = x*K; %幅值输出
end
angle_out = radian_out*180/pi; %相角输出</span>
1.2 Verilog代码
采用16级流水的Verilog (未考虑象限问题)
//*********************************************************
//功能1：已知角度θ，求正弦sinθ和余弦cosθ
//思想:若向量模值为1，则其x坐标就是余弦值，y坐标就是正弦值。
//利用这一点，从(K,0)处迭代旋转至θ处的单位矢量即可。
//*********************************************************
module cordic_A(
input clk,
input rst_n,
input [8:0] angle,
input start,
output reg signed[31:0] Sin,
output reg signed[31:0] Cos,
output finished
);
parameter angle_0 = 32'd2949120; //45度*2^16
parameter angle_1 = 32'd1740992; //26.5651度*2^16
parameter angle_2 = 32'd919872; //14.0362度*2^16
parameter angle_3 = 32'd466944; //7.1250度*2^16
parameter angle_4 = 32'd234368; //3.5763度*2^16
parameter angle_5 = 32'd117312; //1.7899度*2^16
parameter angle_6 = 32'd58688; //0.8952度*2^16
parameter angle_7 = 32'd29312; //0.4476度*2^16
parameter angle_8 = 32'd14656; //0.2238度*2^16
parameter angle_9 = 32'd7360; //0.1119度*2^16
parameter angle_10 = 32'd3648; //0.0560度*2^16
parameter angle_11 = 32'd1856; //0.0280度*2^16
parameter angle_12 = 32'd896; //0.0140度*2^16
parameter angle_13 = 32'd448; //0.0070度*2^16
parameter angle_14 = 32'd256; //0.0035度*2^16
parameter angle_15 = 32'd128; //0.0018度*2^16
parameter pipeline = 16;
parameter K = 32'h09b74; //0.607253*2^16,
// reg signed [31:0] Sin;
// reg signed [31:0] Cos;
reg signed [31:0] x0 =0,y0 =0,z0 =0;
reg signed [31:0] x1 =0,y1 =0,z1 =0;
reg signed [31:0] x2 =0,y2 =0,z2 =0;
reg signed [31:0] x3 =0,y3 =0,z3 =0;
reg signed [31:0] x4 =0,y4 =0,z4 =0;
reg signed [31:0] x5 =0,y5 =0,z5 =0;
reg signed [31:0] x6 =0,y6 =0,z6 =0;
reg signed [31:0] x7 =0,y7 =0,z7 =0;
reg signed [31:0] x8 =0,y8 =0,z8 =0;
reg signed [31:0] x9 =0,y9 =0,z9 =0;
reg signed [31:0] x10=0,y10=0,z10=0;
reg signed [31:0] x11=0,y11=0,z11=0;
reg signed [31:0] x12=0,y12=0,z12=0;
reg signed [31:0] x13=0,y13=0,z13=0;
reg signed [31:0] x14=0,y14=0,z14=0;
reg signed [31:0] x15=0,y15=0,z15=0;
reg signed [31:0] x16=0,y16=0,z16=0;
reg [4:0] count;
always@(posedge clk or negedge rst_n)begin
if(!rst_n)
count <= 'b0;
else if(start)begin
if(count != 5'd18)
count <= count + 1'b1;
else
count <= count;
end
end
assign finished = (count == 5'd18)?1'b1:1'b0;
always@(posedge clk or negedge rst_n)begin
if(!rst_n)begin
x0 <= 'b0;
y0 <= 'b0;
z0 <= 'b0;
end
else begin
x0 <= K;
y0 <= 32'd0;
z0 <= angle << 16;
end
end
always@(posedge clk or negedge rst_n)begin//第一次迭代
if(!rst_n)begin
x1 <= 'b0;
y1 <= 'b0;
z1 <= 'b0;
end
else if(z0[31]) begin
x1 <= x0 + y0;
y1 <= y0 - x0;
z1 <= z0 + angle_0;
end
else begin
x1 <= x0 - y0;
y1 <= y0 + x0;
z1 <= z0 - angle_0;
end
end
always@(posedge clk or negedge rst_n)begin//第二次迭代
if(!rst_n)begin
x2 <= 'b0;
y2 <= 'b0;
z2 <= 'b0;
end
else if(z1[31]) begin
x2 <= x1 + (y1>>>1);
y2 <= y1 - (x1>>>1);
z2 <= z1 + angle_1;
end
else begin
x2 <= x1 - (y1>>>1);
y2 <= y1 + (x1>>>1);
z2 <= z1 - angle_1;
end
end
always@(posedge clk or negedge rst_n)begin//第3次迭代
if(!rst_n)begin
x3 <= 'b0;
y3 <= 'b0;
z3 <= 'b0;
end
else if(z2[31]) begin
x3 <= x2 + (y2>>>2);
y3 <= y2 - (x2>>>2);
z3 <= z2 + angle_2;
end
else begin
x3 <= x2 - (y2>>>2);
y3 <= y2 + (x2>>>2);
z3 <= z2 - angle_2;
end
end
always@(posedge clk or negedge rst_n)begin//第4次迭代
if(!rst_n)begin
x4 <= 'b0;
y4 <= 'b0;
z4 <= 'b0;
end
else if(z3[31]) begin
x4 <= x3 + (y3>>>3);
y4 <= y3 - (x3>>>3);
z4 <= z3 + angle_3;
end
else begin
x4 <= x3 - (y3>>>3);
y4 <= y3 + (x3>>>3);
z4 <= z3 - angle_3;
end
end
always@(posedge clk or negedge rst_n)begin//第5次迭代
if(!rst_n)begin
x5 <= 'b0;
y5 <= 'b0;
z5 <= 'b0;
end
else if(z4[31]) begin
x5 <= x4 + (y4>>>4);
y5 <= y4 - (x4>>>4);
z5 <= z4 + angle_4;
end
else begin
x5 <= x4 - (y4>>>4);
y5 <= y4 + (x4>>>4);
z5 <= z4 - angle_4;
end
end
always@(posedge clk or negedge rst_n)begin//第6次迭代
if(!rst_n)begin
x6 <= 'b0;
y6 <= 'b0;
z6 <= 'b0;
end
else if(z5[31]) begin
x6 <= x5 + (y5>>>5);
y6 <= y5 - (x5>>>5);
z6 <= z5 + angle_5;
end
else begin
x6 <= x5 - (y5>>>5);
y6 <= y5 + (x5>>>5);
z6 <= z5 - angle_5;
end
end
always@(posedge clk or negedge rst_n)begin//第7次迭代
if(!rst_n)begin
x7 <= 'b0;
y7 <= 'b0;
z7 <= 'b0;
end
else if(z6[31]) begin
x7 <= x6 + (y6>>>6);
y7 <= y6 - (x6>>>6);
z7 <= z6 + angle_6;
end
else begin
x7 <= x6 - (y6>>>6);
y7 <= y6 + (x6>>>6);
z7 <= z6 - angle_6;
end
end
always@(posedge clk or negedge rst_n)begin//第8次迭代
if(!rst_n)begin
x8 <= 'b0;
y8 <= 'b0;
z8 <= 'b0;
end
else if(z7[31]) begin
x8 <= x7 + (y7>>>7);
y8 <= y7 - (x7>>>7);
z8 <= z7 + angle_7;
end
else begin
x8 <= x7 - (y7>>>7);
y8 <= y7 + (x7>>>7);
z8 <= z7 - angle_7;
end
end
always@(posedge clk or negedge rst_n)begin//第9次迭代
if(!rst_n)begin
x9 <= 'b0;
y9 <= 'b0;
z9 <= 'b0;
end
else if(z8[31]) begin
x9 <= x8 + (y8>>>8);
y9 <= y8 - (x8>>>8);
z9 <= z8 + angle_8;
end
else begin
x9 <= x8 - (y8>>>8);
y9 <= y8 + (x8>>>8);
z9 <= z8 - angle_8;
end
end
always@(posedge clk or negedge rst_n)begin//第10次迭代
if(!rst_n)begin
x10 <= 'b0;
y10 <= 'b0;
z10 <= 'b0;
end
else if(z9[31]) begin
x10 <= x9 + (y9>>>9);
y10 <= y9 - (x9>>>9);
z10 <= z9 + angle_9;
end
else begin
x10 <= x9 - (y9>>>9);
y10 <= y9 + (x9>>>9);
z10 <= z9 - angle_9;
end
end
always@(posedge clk or negedge rst_n)begin//第11次迭代
if(!rst_n)begin
x11 <= 'b0;
y11 <= 'b0;
z11 <= 'b0;
end
else if(z10[31]) begin
x11 <= x10 + (y10>>>10);
y11 <= y10 - (x10>>>10);
z11 <= z10 + angle_10;
end
else begin
x11 <= x10 - (y10>>>10);
y11 <= y10 + (x10>>>10);
z11 <= z10 - angle_10;
end
end
always@(posedge clk or negedge rst_n)begin//第12次迭代
if(!rst_n)begin
x12 <= 'b0;
y12 <= 'b0;
z12 <= 'b0;
end
else if(z11[31]) begin
x12 <= x11 + (y11>>>11);
y12 <= y11 - (x11>>>11);
z12 <= z11 + angle_11;
end
else begin
x12 <= x11 - (y11>>>11);
y12 <= y11 + (x11>>>11);
z12 <= z11 - angle_11;
end
end
always@(posedge clk or negedge rst_n)begin//第13次迭代
if(!rst_n)begin
x13 <= 'b0;
y13 <= 'b0;
z13 <= 'b0;
end
else if(z12[31]) begin
x13 <= x12 + (y12>>>12);
y13 <= y12 - (x12>>>12);
z13 <= z12 + angle_12;
end
else begin
x13 <= x12 - (y12>>>12);
y13 <= y12 + (x12>>>12);
z13 <= z12 - angle_12;
end
end
always@(posedge clk or negedge rst_n)begin//第14次迭代
if(!rst_n)begin
x14 <= 'b0;
y14 <= 'b0;
z14 <= 'b0;
end
else if(z13[31]) begin
x14 <= x13 + (y13>>>13);
y14 <= y13 - (x13>>>13);
z14 <= z13 + angle_13;
end
else begin
x14 <= x13 - (y13>>>13);
y14 <= y13 + (x13>>>13);
z14 <= z13 - angle_13;
end
end
always@(posedge clk or negedge rst_n)begin//第15次迭代
if(!rst_n)begin
x15 <= 'b0;
y15 <= 'b0;
z15 <= 'b0;
end
else if(z14[31]) begin
x15 <= x14 + (y14>>>14);
y15 <= y14 - (x14>>>14);
z15 <= z14 + angle_14;
end
else begin
x15 <= x14 - (y14>>>14);
y15 <= y14 + (x14>>>14);
z15 <= z14 - angle_14;
end
end
always@(posedge clk or negedge rst_n)begin//第16次迭代
if(!rst_n)begin
x16 <= 'b0;
y16 <= 'b0;
z16 <= 'b0;
end
else if(z15[31]) begin
x16 <= x15 + (y15>>>15);
y16 <= y15 - (x15>>>15);
z16 <= z15 + angle_15;
end
else begin
x16 <= x15 - (y15>>>15);
y16 <= y15 + (x15>>>15);
z16 <= z15 - angle_15;
end
end
always@(posedge clk or negedge rst_n)begin
if(!rst_n)begin
Cos <= 'b0;
Sin <= 'b0;
end
else begin
Sin = y16;
Cos = x16;
end
end
endmodule
1.3 testbench A
输入连续5个时钟上升沿输入60°，30°，45°，90°，30°，看是否能够连续输出相应结果。
`timescale 1ns/1ps
module cordic_Atb();
parameter PERIOD = 10;
reg clk;
reg rst_n;
reg [8:0]angle;
reg start;
wire [31:0] Sin;
wire [31:0] Cos;
wire finished;
initial begin
clk = 0;
rst_n = 0;
start = 0;
angle = 'b0;
#100 rst_n =1;
#10 @(posedge clk) start = 1'b1;angle = 9'd60;
#10 @(posedge clk) angle = 9'd30;
#10 @(posedge clk) angle = 9'd45;
#10 @(posedge clk) angle = 9'd90;
#10 @(posedge clk) angle = 9'd30;
#100000 $stop;
end
always #(PERIOD/2) clk = ~clk;
cordic_A inst1(
.clk(clk),
.rst_n(rst_n),
.angle(angle),
.start(start),
.Sin(Sin),
.Cos(Cos),
.finished(finished)
);
endmodule
相应仿真波形如图所示：
由于硬件实现将数据整体左移16位，从而换区更高精度，因此得到的结果需要向右移16位，进行还原。还原后我们发现跟预期的值相等。
2. 已 知 角 度 θ ， 求 正 弦 s i n θ 和 余 弦 c o s θ \color{#F00}{2.已知角度θ，求正弦sinθ和余弦cosθ} 2.已知角度θ，求正弦sinθ和余弦cosθ
思想:若向量模值为1，则其x坐标就是余弦值，y坐标就是正弦值。利用这一点，从(K,0)处迭代旋转至θ处的单位矢量即可。
迭代方程及K的计算同第一小节。同时也要注意预先对象限的判断和补偿。
2.1 MATLAB代码
%% ***********************************************************************************
% 已知相角theta，计算其正弦和余弦值。基本公式如下：
% x(k+1) = x(k) - d(k)*y(k)*2^(-k)
% y(k+1) = y(k) + d(k)*x(k)*2^(-k)
% z(k) = z(k) - d(k)*actan(2^(-k))
%% ***********************************************************************************
clear;close all;clc;
% 初始化----------------------------------------
N = 16; %迭代次数
tan_table = 2.^-(0 : N-1);
angle_LUT = atan(tan_table);
K = 1;
for k = 0 : N-1
K = K*(1/sqrt(1 + 2^(-2*k)));
end
theta = -90;
x = 1;
y = 0;
phase_accumulate = theta/180*pi; %转化为弧度
% cordic算法计算-------------------------------
if (phase_accumulate > pi/2) % 先做象限判断，得到相位补偿值
phase_accumulate = phase_accumulate - pi;
sign_x = -1;
sign_y = -1;
elseif (phase_accumulate < -pi/2)
phase_accumulate = phase_accumulate + pi;
sign_x = -1;
sign_y = -1;
else
sign_x = 1;
sign_y = 1;
end
for k = 0 : N-1 % 迭代开始
x_temp = x;
if (phase_accumulate > 0) % d(k)=1，逆时针旋转
x = x_temp - y*2^(-k);
y = y + x_temp*2^(-k);
phase_accumulate = phase_accumulate - angle_LUT(k+1);
else % d(k)=-1，顺时针旋转
x = x_temp + y*2^(-k);
y = y - x_temp*2^(-k);
phase_accumulate = phase_accumulate + angle_LUT(k+1);
end
end
cos_out = sign_x*x*K; %余弦输出
sin_out = sign_y*y*K; %正弦输出</span>
2.2 Verilog代码
16级流水的Verilog代码如下：
//*********************************************************
//已知坐标(x,y)，求其向量对应的相角θ（反正切）和模值
//思想:把原向量逐次向X正轴进行伪旋转逼近.
//整个过程的累计旋转角度即为θ，
//而旋转后的x坐标补偿模值增益K后即为原向量模值。
//*********************************************************
module cordic_B(
input clk,
input rst_n,
input signed [31:0] x,
input signed [31:0] y,
input start,
output reg signed[31:0] angle,
output reg signed[31:0] mozhi,
output finished
);
parameter angle_0 = 32'd2949120; //45度*2^16
parameter angle_1 = 32'd1740992; //26.5651度*2^16
parameter angle_2 = 32'd919872; //14.0362度*2^16
parameter angle_3 = 32'd466944; //7.1250度*2^16
parameter angle_4 = 32'd234368; //3.5763度*2^16
parameter angle_5 = 32'd117312; //1.7899度*2^16
parameter angle_6 = 32'd58688; //0.8952度*2^16
parameter angle_7 = 32'd29312; //0.4476度*2^16
parameter angle_8 = 32'd14656; //0.2238度*2^16
parameter angle_9 = 32'd7360; //0.1119度*2^16
parameter angle_10 = 32'd3648; //0.0560度*2^16
parameter angle_11 = 32'd1856; //0.0280度*2^16
parameter angle_12 = 32'd896; //0.0140度*2^16
parameter angle_13 = 32'd448; //0.0070度*2^16
parameter angle_14 = 32'd256; //0.0035度*2^16
parameter angle_15 = 32'd128; //0.0018度*2^16
parameter pipeline = 16;
parameter K = 32'h09b74; //0.607253*2^16,
// reg signed [31:0] Sin;
// reg signed [31:0] Cos;
reg signed [31:0] x0 =0,y0 =0,z0 =0;
reg signed [31:0] x1 =0,y1 =0,z1 =0;
reg signed [31:0] x2 =0,y2 =0,z2 =0;
reg signed [31:0] x3 =0,y3 =0,z3 =0;
reg signed [31:0] x4 =0,y4 =0,z4 =0;
reg signed [31:0] x5 =0,y5 =0,z5 =0;
reg signed [31:0] x6 =0,y6 =0,z6 =0;
reg signed [31:0] x7 =0,y7 =0,z7 =0;
reg signed [31:0] x8 =0,y8 =0,z8 =0;
reg signed [31:0] x9 =0,y9 =0,z9 =0;
reg signed [31:0] x10=0,y10=0,z10=0;
reg signed [31:0] x11=0,y11=0,z11=0;
reg signed [31:0] x12=0,y12=0,z12=0;
reg signed [31:0] x13=0,y13=0,z13=0;
reg signed [31:0] x14=0,y14=0,z14=0;
reg signed [31:0] x15=0,y15=0,z15=0;
reg signed [31:0] x16=0,y16=0,z16=0;
reg [4:0] count;
always@(posedge clk or negedge rst_n)begin
if(!rst_n)
count <= 'b0;
else if(start)begin
if(count != 5'd18)
count <= count + 1'b1;
else
count <= count;
end
end
assign finished = (count == 5'd18)?1'b1:1'b0;
always@(posedge clk or negedge rst_n)begin
if(!rst_n)begin
x0 <= 'b0;
y0 <= 'b0;
z0 <= 'b0;
end
else begin
x0 <= x<<<16;
y0 <= y<<<16;
z0 <= 'b0;
end
end
always@(posedge clk or negedge rst_n)begin//第一次迭代
if(!rst_n)begin
x1 <= 'b0;
y1 <= 'b0;
z1 <= 'b0;
end
else if(!y0[31]) begin //和A不同的是，每次判断y坐标的正负，从而决定是顺时针还是逆时针旋转。
x1 <= x0 + y0;
y1 <= y0 - x0;
z1 <= z0 + angle_0;
end
else begin
x1 <= x0 - y0;
y1 <= y0 + x0;
z1 <= z0 - angle_0;
end
end
always@(posedge clk or negedge rst_n)begin//第二次迭代
if(!rst_n)begin
x2 <= 'b0;
y2 <= 'b0;
z2 <= 'b0;
end
else if(!y1[31]) begin
x2 <= x1 + (y1>>>1);
y2 <= y1 - (x1>>>1);
z2 <= z1 + angle_1;
end
else begin
x2 <= x1 - (y1>>>1);
y2 <= y1 + (x1>>>1);
z2 <= z1 - angle_1;
end
end
always@(posedge clk or negedge rst_n)begin//第3次迭代
if(!rst_n)begin
x3 <= 'b0;
y3 <= 'b0;
z3 <= 'b0;
end
else if(!y2[31]) begin
x3 <= x2 + (y2>>>2);
y3 <= y2 - (x2>>>2);
z3 <= z2 + angle_2;
end
else begin
x3 <= x2 - (y2>>>2);
y3 <= y2 + (x2>>>2);
z3 <= z2 - angle_2;
end
end
always@(posedge clk or negedge rst_n)begin//第4次迭代
if(!rst_n)begin
x4 <= 'b0;
y4 <= 'b0;
z4 <= 'b0;
end
else if(!y3[31]) begin
x4 <= x3 + (y3>>>3);
y4 <= y3 - (x3>>>3);
z4 <= z3 + angle_3;
end
else begin
x4 <= x3 - (y3>>>3);
y4 <= y3 + (x3>>>3);
z4 <= z3 - angle_3;
end
end
always@(posedge clk or negedge rst_n)begin//第5次迭代
if(!rst_n)begin
x5 <= 'b0;
y5 <= 'b0;
z5 <= 'b0;
end
else if(!y4[31]) begin
x5 <= x4 + (y4>>>4);
y5 <= y4 - (x4>>>4);
z5 <= z4 + angle_4;
end
else begin
x5 <= x4 - (y4>>>4);
y5 <= y4 + (x4>>>4);
z5 <= z4 - angle_4;
end
end
always@(posedge clk or negedge rst_n)begin//第6次迭代
if(!rst_n)begin
x6 <= 'b0;
y6 <= 'b0;
z6 <= 'b0;
end
else if(!y5[31]) begin
x6 <= x5 + (y5>>>5);
y6 <= y5 - (x5>>>5);
z6 <= z5 + angle_5;
end
else begin
x6 <= x5 - (y5>>>5);
y6 <= y5 + (x5>>>5);
z6 <= z5 - angle_5;
end
end
always@(posedge clk or negedge rst_n)begin//第7次迭代
if(!rst_n)begin
x7 <= 'b0;
y7 <= 'b0;
z7 <= 'b0;
end
else if(!y6[31]) begin
x7 <= x6 + (y6>>>6);
y7 <= y6 - (x6>>>6);
z7 <= z6 + angle_6;
end
else begin
x7 <= x6 - (y6>>>6);
y7 <= y6 + (x6>>>6);
z7 <= z6 - angle_6;
end
end
always@(posedge clk or negedge rst_n)begin//第8次迭代
if(!rst_n)begin
x8 <= 'b0;
y8 <= 'b0;
z8 <= 'b0;
end
else if(!y7[31]) begin
x8 <= x7 + (y7>>>7);
y8 <= y7 - (x7>>>7);
z8 <= z7 + angle_7;
end
else begin
x8 <= x7 - (y7>>>7);
y8 <= y7 + (x7>>>7);
z8 <= z7 - angle_7;
end
end
always@(posedge clk or negedge rst_n)begin//第9次迭代
if(!rst_n)begin
x9 <= 'b0;
y9 <= 'b0;
z9 <= 'b0;
end
else if(!y8[31]) begin
x9 <= x8 + (y8>>>8);
y9 <= y8 - (x8>>>8);
z9 <= z8 + angle_8;
end
else begin
x9 <= x8 - (y8>>>8);
y9 <= y8 + (x8>>>8);
z9 <= z8 - angle_8;
end
end
always@(posedge clk or negedge rst_n)begin//第10次迭代
if(!rst_n)begin
x10 <= 'b0;
y10 <= 'b0;
z10 <= 'b0;
end
else if(!y9[31]) begin
x10 <= x9 + (y9>>>9);
y10 <= y9 - (x9>>>9);
z10 <= z9 + angle_9;
end
else begin
x10 <= x9 - (y9>>>9);
y10 <= y9 + (x9>>>9);
z10 <= z9 - angle_9;
end
end
always@(posedge clk or negedge rst_n)begin//第11次迭代
if(!rst_n)begin
x11 <= 'b0;
y11 <= 'b0;
z11 <= 'b0;
end
else if(!y10[31]) begin
x11 <= x10 + (y10>>>10);
y11 <= y10 - (x10>>>10);
z11 <= z10 + angle_10;
end
else begin
x11 <= x10 - (y10>>>10);
y11 <= y10 + (x10>>>10);
z11 <= z10 - angle_10;
end
end
always@(posedge clk or negedge rst_n)begin//第12次迭代
if(!rst_n)begin
x12 <= 'b0;
y12 <= 'b0;
z12 <= 'b0;
end
else if(!y11[31]) begin
x12 <= x11 + (y11>>>11);
y12 <= y11 - (x11>>>11);
z12 <= z11 + angle_11;
end
else begin
x12 <= x11 - (y11>>>11);
y12 <= y11 + (x11>>>11);
z12 <= z11 - angle_11;
end
end
always@(posedge clk or negedge rst_n)begin//第13次迭代
if(!rst_n)begin
x13 <= 'b0;
y13 <= 'b0;
z13 <= 'b0;
end
else if(!y12[31]) begin
x13 <= x12 + (y12>>>12);
y13 <= y12 - (x12>>>12);
z13 <= z12 + angle_12;
end
else begin
x13 <= x12 - (y12>>>12);
y13 <= y12 + (x12>>>12);
z13 <= z12 - angle_12;
end
end
always@(posedge clk or negedge rst_n)begin//第14次迭代
if(!rst_n)begin
x14 <= 'b0;
y14 <= 'b0;
z14 <= 'b0;
end
else if(!y13[31]) begin
x14 <= x13 + (y13>>>13);
y14 <= y13 - (x13>>>13);
z14 <= z13 + angle_13;
end
else begin
x14 <= x13 - (y13>>>13);
y14 <= y13 + (x13>>>13);
z14 <= z13 - angle_13;
end
end
always@(posedge clk or negedge rst_n)begin//第15次迭代
if(!rst_n)begin
x15 <= 'b0;
y15 <= 'b0;
z15 <= 'b0;
end
else if(!y14[31]) begin
x15 <= x14 + (y14>>>14);
y15 <= y14 - (x14>>>14);
z15 <= z14 + angle_14;
end
else begin
x15 <= x14 - (y14>>>14);
y15 <= y14 + (x14>>>14);
z15 <= z14 - angle_14;
end
end
always@(posedge clk or negedge rst_n)begin//第16次迭代
if(!rst_n)begin
x16 <= 'b0;
y16 <= 'b0;
z16 <= 'b0;
end
else if(!y15[31]) begin
x16 <= x15 + (y15>>>15);
y16 <= y15 - (x15>>>15);
z16 <= z15 + angle_15;
end
else begin
x16 <= x15 - (y15>>>15);
y16 <= y15 + (x15>>>15);
z16 <= z15 - angle_15;
end
end
wire[31:0] mozhi_tmp;
wire[31:0] mozhi_tmp1;
wire[31:0] mozhi_tmp2;
assign mozhi_tmp = x16>>>6;
assign mozhi_tmp1 = mozhi_tmp+mozhi_tmp+mozhi_tmp+mozhi_tmp+mozhi_tmp+mozhi_tmp+mozhi_tmp;
assign mozhi_tmp2 = x16>>>1;
always@(posedge clk or negedge rst_n)begin
if(!rst_n)begin
mozhi <= 'b0;
angle <= 'b0;
end
else begin
mozhi = (mozhi_tmp1 + mozhi_tmp2)>>>16;
angle = z16>>>16;
end
end
endmodule
2.3 Testbench B
Testbench代码如下：
`timescale 1ns/1ps
module cordic_Btb();
parameter PERIOD = 10;
reg clk;
reg rst_n;
reg signed [31:0] x;
reg signed [31:0] y;
reg start;
wire signed [31:0] angle;
wire signed [31:0] mozhi;
wire finished;
initial begin
clk = 0;
rst_n = 0;
x = 'b0;
y = 'b0;
start = 0;
#100 rst_n = 1;
#10 @(posedge clk) start = 1; x = 9'd100 ; y = 9'd100 ;
#10 @(posedge clk) x = 9'd10; y = 9'd10;
#10 @(posedge clk) x = 9'd3 ; y = 9'd4 ;
#10 @(posedge clk) x = 9'd6 ; y = 9'd8 ;
#100000 $stop;
end
always #(PERIOD/2) clk = ~clk;
cordic_B inst1(
.clk(clk),
.rst_n(rst_n),
.x(x),
.y(y),
.start(start),
.angle(angle),
.mozhi(mozhi),
.finished(finished)
);
endmodule
相应仿真波形如下图所示：
可以看到，输入的(x,y)为(100,100)时，其夹角 θ \theta θ=45°，模值近似为141。其他情况同理。
| END |
|---|
*本文的Verilog代码均没有考虑象限问题，MATLAB代码有考虑。
*本文参考博客
[1]https://blog.csdn.net/qq_39210023/article/details/77456031
[2]https://blog.csdn.net/longxuekun1992/article/details/52435024